A Proof of Ceperley's Conjecture on the Nodal Maximality of Ground States

Even in the absence of a magnetic field, any eigenstate of a system of many identical fermions is necessarily signed and assumes both positive and negative values. A long-standing conjecture by Ceperley posits that the positive node of the ground state is maximal, namely, the supports of the positive part of its wavefunction are topologically connected into a single node. Such nodal maximality is theoretically interesting and practically important for quantum Monte Carlo. It has been proved true recently for a system of non-interacting fermions in a multi-dimensional space. However, the problem remains open for a general Schrödinger operator with interactions among identical fermions.

Reported here is a proof of Ceperley's conjecture in full generality and mathematical rigor, which is inspired by the physics of adiabatic evolution. It uses an analytic curve to evolve an interaction-free Hamiltonian to a general Hamiltonian with interactions, then employs Rellich's theorem on the analyticity of the curves of eigen solutions and the Hopf lemma on the nonvanishingness of the boundary normal derivative of any eigen function of an elliptic operator to assert that a positive node starting maximal must remain maximal throughout the analytic curve of Schrödinger operators, provided that the ground states remain nondegenerate along the curve. The proof is completed by showing that any ground state degeneracy can be removed by introducing a perturbative interaction.